#### Geometry & Topology Monographs 2 (1999),
Proceedings of the Kirbyfest,
paper no. 4, pages 87-102.

## Topological Field Theories and formulae of Casson and Meng-Taubes

### S K Donaldson

**Abstract**.
The goal of this paper is to give a new proof of a theorem of Meng and
Taubes that identifies the Seiberg-Witten invariants of 3-manifolds
with Milnor torsion. The point of view here will be that of
topological quantum field theory. In particular, we relate the
Seiberg-Witten equations on a 3-manifold with the Abelian vortex
equations on a Riemann surface. These techniques also give a new proof
of the surgery formula for the Casson invariant, interpreted as an
invariant of a homology S^2 x S^1.
**Keywords**.
Seiberg-Witten invariant, Casson invariant, Alexander polynomial, Milnor torsion, topological quantum field theory, moduli space, vortex equation

**AMS subject classification**.
Primary: 57R57. Secondary: 57M25, 57N10, 58D29.

**E-print:** `arXiv:math.GT/9911248`

Submitted: 5 March 1999.
(Revised: 24 June 1999.)
Published: 17 November 1999.

Notes on file formats
S K Donaldson

Department of Mathematics

Imperial College, London SW7 2BZ, UK

Email: s.donaldson@ic.ac.uk

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